To prove that [tex]\(\tan \alpha \cdot \cot \alpha=1\)[/tex], let's break it down step-by-step.
1. Understand the Definitions:
- The tangent function is defined as:
[tex]\[
\tan \alpha = \frac{\sin \alpha}{\cos \alpha}
\][/tex]
- The cotangent function is the reciprocal of the tangent function, defined as:
[tex]\[
\cot \alpha = \frac{1}{\tan \alpha} = \frac{\cos \alpha}{\sin \alpha}
\][/tex]
2. Multiply the Two Functions:
When we multiply [tex]\(\tan \alpha\)[/tex] by [tex]\(\cot \alpha\)[/tex], we get:
[tex]\[
\tan \alpha \cdot \cot \alpha = \left(\frac{\sin \alpha}{\cos \alpha}\right) \cdot \left(\frac{\cos \alpha}{\sin \alpha}\right)
\][/tex]
3. Simplify the Expression:
When multiplying these fractions, the numerator of the first fraction ([tex]\(\sin \alpha\)[/tex]) gets multiplied by the numerator of the second fraction ([tex]\(\cos \alpha\)[/tex]), and the denominator of the first fraction ([tex]\(\cos \alpha\)[/tex]) gets multiplied by the denominator of the second fraction ([tex]\(\sin \alpha\)[/tex]). This results in:
[tex]\[
\tan \alpha \cdot \cot \alpha = \frac{\sin \alpha \cdot \cos \alpha}{\cos \alpha \cdot \sin \alpha}
\][/tex]
4. Cancel Out the Common Terms:
Both the numerator and the denominator are identical:
[tex]\[
\frac{\sin \alpha \cdot \cos \alpha}{\cos \alpha \cdot \sin \alpha}
\][/tex]
When we divide a term by itself, it equals 1:
[tex]\[
\tan \alpha \cdot \cot \alpha = 1
\][/tex]
Thus, we have proven that:
[tex]\[
\tan \alpha \cdot \cot \alpha = 1
\][/tex]
